Understanding axis locking and local coordinate systems
You haven't been helped by a bit of outdated documentation (reported), or, IMO, by the interface.
First of all,though, the Local: orientation is the orientation of the Object axes. These can be displayed by checking Axes the Object tab &gt; Viewport Display panel. Unfortunately the display is not 'In Front', so you may have to AltZ switch on X-ray mode, to see them. The Object is a vehicle for its mesh / curve elements, and carries all the object-level transforms.
Normal orientation reverts to Local in Object mode. In Edit mode, actually, it's not always predictable for modeling. Vertex normals may be calculated from adjacent faces, if there, or by projection from the object origin, if not. The useful, predictable normals, I think, are: from selected faces: Z will be along their mean normal. From 2 vertices or an edge, Y will be along the line between the verts. From 3 vertices, Z will be normal to the plane defined by them.
Contrary to your quote from the manual, a single tap on X, Y or Z will constrain a transform to the orientation displayed in the Transform Orientation dropdown. The second tap will revert to Global, (or from Global to Local, if Global is already selected). The third tap will release the constraint.
Custom Orientation
Following the above Normal rules, you can create your own orientation from a selection of elements by hitting the little '+' in the Orientation dropdown.
Extrude
Just to make everything more confusing, E Extrude behaves differently. You're right. The Normal orientation is assumed as the background behaviour, taking the place of Global in the transform tap-sequence described above, and the tool behaves as if you had tapped ZZ already, so the next Z releases, and the next puts you into the orientation in the dropdown.
If, in an extrude, you tap a constraint other than Z, the tool assumes you are starting the cycle from the beginning: Dropdown &gt; Normal &gt; Free.
The old behaviour of the constraints was as in your quote from the manual. It really seems to me that flipping the single and double-tap behaviour has opened up a can of worms. Personally I would rather have had consistency than 'convenience'.